degree. Moreover, by Remark 4.1 each C i is contained in Y = { F = y 1 f 1 + ⋯ + y k f k = 0 } and\n\nfurthermore it has codimension k .\n\nClaim: { C i } ni = 1 is a basis of prim ( ) . It is enough to prove that λ C i is diﬀerent from zero in H k,k prim ( Y, Q ) or equivalently that the cohomology classes { λ C i } ni = 1 do not come from the ambient space. By contradiction, let us assume that there exists a j and C ⊂ P 2 k + 1 Σ ,X such that λ C ∈ H k,k ( P 2 k + 1 Σ ,X , Q ) with i ∗ ( λ C ) = λ C j or in terms of homology there exists a ( k + 2 ) -dimensional algebraic subvariety V ⊂ P 2 k + 1 Σ ,X such that V ∩ Y = C j so